Radiation Safety and Logarithms

If a site is contaminated with a
radioactive material of half life one week and the level of radiation is now
ten times the safe level, how long will it be before it is safe?

Using the formula with where is the safe level, we need to find t giving ,
i.e. such that .

So we need to find the exponent t in the equation .

Clearly t=3
is not enough, since after 3 weeks the radiation level will have been
multiplied by and so .
But by t=4 the radiation will be
below the safe level, so the solution is somewhere between 3 and 4.

If we want to find exactly when the safe
level is reached then we need to solve the equation exactly.

Since we are solving to find an exponent,
this is called an exponential equation.